Backward martingale transport and Fitzpatrick functions in pseudo-Euclidean spaces

We study an optimal transport problem with a backward martingale constraint in a pseudo-Euclidean space $S$. We show that the dual problem consists in the minimization of the expected values of the Fitzpatrick functions associated with maximal $S$-monotone sets. An optimal plan $\gamma$ and an optimal maximal $S$-monotone set $G$ are characterized by the condition that the support of $\gamma$ is contained in the graph of the $S$-projection on $G$. For a Gaussian random variable $Y$, we get a unique decomposition: $Y = X+Z$, where $X$ and $Z$ are independent Gaussian random variables taking values, respectively, in complementary positive and negative linear subspaces of the $S$-space.